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   "source": [
    "# High-order polynomial approximations to special functions and integral transforms"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The aim of this example is to show that using `TaylorIntegration.jl` we can evaluate with high accuracy definite integrals of the form \n",
    "\n",
    "$$\n",
    "I_{\\omega}(x) = \\int_{x_0}^xf_{\\omega}(t)\\mathrm{d} t\n",
    "$$\n",
    "\n",
    "for smooth integrands $f$ which might depend on a parameter $\\omega$. In particular, combining high-acurracy integration with jet transport techniques, we are able to obtain high-order, dense polynomial representations of special functions and integral transforms. The trick is to translate the definite integral problem to an initial value problem for an explicit ODE, thanks to the fundamental theorem of calculus."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Error function: $\\operatorname{erf}(x)=\\frac{2}{\\sqrt{\\pi}}\\int_0^x \\exp(-t^2) dt$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a first example, we will evaluate the [__error function__](https://en.wikipedia.org/wiki/Error_function), defined by the integral $\\operatorname{erf}(x)=\\frac{2}{\\sqrt{\\pi}}\\int_0^x \\exp(-t^2) dt$ using `TaylorIntegration.jl`. We proceed as follows:\n",
    "\n",
    "First, define a suitable ODE system equivalent to the definite integral problem at hand, and then integrate it. Indeed, by the fundamental theorem of calculus we have\n",
    "\n",
    "$$\n",
    "\\operatorname{erf} '(x) =\\frac{2}{\\sqrt{\\pi}}\\exp(-x^2)\n",
    "$$\n",
    "\n",
    "\n",
    "where the prime $'$ denotes differentiation with respect to $x$. Therefore, we reinterpret $\\operatorname{erf}(x)$ as the solution of the initial value problem defined by the ODE\n",
    "\n",
    "$$\n",
    "\\frac{d\\operatorname{erf}}{dx} = \\frac{2}{\\sqrt{\\pi}}\\exp(-x^2)\n",
    "$$\n",
    "\n",
    "subject to the initial condition $\\operatorname{erf}(0)=0$. We will integrate this ODE in order to evaluate $\\operatorname{erf}(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by loading `TaylorIntegration` and `Plots` with the `GR` backend:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using TaylorIntegration, Plots\n",
    "gr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the RHS of the ODE we want to solve (below, `t` is the independent variable):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(t, x) = (2/sqrt(pi))*exp(-t^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameters we will use for the Taylor integration are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0e-20"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = 0.0 #the initial value of the independent variable; in this case, x\n",
    "xmax = 10.0 #the final value of the independent variable; in this case, x\n",
    "erf0 = 0.0 #the initial condition for erf: erf(0) = 0\n",
    "const order = 25 #the order of the Taylor expansions\n",
    "const abstol = 1e-20 #the absolute local error tolerance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial conditions are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 0.0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0, erf0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we proceed to integrate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.335947 seconds (147.59 k allocations: 7.766 MiB)\n"
     ]
    }
   ],
   "source": [
    "@time xv, erfv = taylorinteg(f, erf0, x0, xmax, order, abstol, maxsteps=5000);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The final $x$ and $\\operatorname{erf}$ values are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10.0, 1.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xv[end], erfv[end]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, how does $\\operatorname{erf}(x)$ look as a function of $x$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(xv, erfv, leg=false)\n",
    "scatter!(xv, erfv, ms=3.0)\n",
    "\n",
    "xlabel!(\"x\")\n",
    "ylabel!(\"erf(x)\")\n",
    "title!(\"Evaluating erf(x) using TaylorIntegration\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "At a first glance, this looks great! But, how do these values actually compare to `SpecialFunctions.jl`'s `erf` implementation?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import SpecialFunctions: erf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "```\n",
       "erf(x)\n",
       "```\n",
       "\n",
       "Compute the error function of `x`, defined by $\\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} dt$ for arbitrary complex `x`.\n"
      ],
      "text/plain": [
       "```\n",
       "erf(x)\n",
       "```\n",
       "\n",
       "Compute the error function of `x`, defined by $\\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} dt$ for arbitrary complex `x`.\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?Base.erf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the absolute difference between our integrated values for $\\operatorname{erf}(x)$ and `Base.erf`, in units of the machine epsilon, as a function of $x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(\n",
    "    xv, (erfv-erf.(xv))/eps(),\n",
    "    xaxis = \"x\",\n",
    "    yaxis = \"Diff: integrated vs Base.erf\",\n",
    "    title = \"Absolute difference between erf(x) and I(x)\",\n",
    "    leg = false\n",
    ")\n",
    "\n",
    "scatter!(xv, (erfv-erf.(xv))/eps(), ms=3.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the error is at most half times `eps()` at each time step!!!!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 2. High-accuracy evaluation of elliptic integrals with `BigFloats`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section, we will evaluate the elliptic integral $K$ using `BigFloats`, up to an accuracy of about $10^{-77}$ (yes, `1E-77`!). It is well known that the complete elliptic integral of the first kind, $K(k)$, may be evaluated via the integral\n",
    "\n",
    "$$\n",
    "K(k) = \\int_0^{\\pi/2}\\frac{ \\mathrm{d}t }{ \\sqrt{1-k^2\\sin^2 t} }\n",
    "$$\n",
    "\n",
    "Again, we can evaluate this integral using `TaylorIntegration.jl`, if we reinterpret this definite integral as the solution of the ODE defined by\n",
    "\n",
    "$$\n",
    "\\frac{\\mathrm{d}K_k}{\\mathrm{d}t} = \\frac{ 1 }{ \\sqrt{1-k^2\\sin^2 t} }\n",
    "$$\n",
    "\n",
    "integrated from $t=0$ to $t=\\pi/2$ for a given parameter $k$, subject to the initial condition $K_k(t=0)=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For definiteness, we will select the parameter value $k_0=0.6051864057360395$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.051864057360394999999999999999999999999999999999999999999999999999999999999984e-01"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k0 = 0.6051864057360395\n",
    "k = big\"0.6051864057360395\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, $k^2$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.662505856877062234277491455602499999999999999999999999999999999999999999999966e-01"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k^2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The RHS for the corresponding ODE may be written as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "G (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G(t,x) = 1/sqrt(1-(k*sin(t))^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we will evaluate this integral using  `BigFloat`s, we will select an order of 90, and a local absolute tolerance near `eps(BigFloat)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.000000000000000000000000000000000000000000000000000000000000000000000000000004e-77"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const bigorder = 90\n",
    "const bigabstol = big\"1.0E-77\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we perform the integration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.561874 seconds (18.57 M allocations: 924.474 MiB, 34.29% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time tv, xv = taylorinteg(G, 0.0, 0.0, BigFloat(π)/2, bigorder, bigabstol);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of $K(k)$ corresponds to the last value of `xv[end]`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.754812577961136589734955441226555912336722052661065250219997312395371169887374"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xv[end]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using `Elliptic.jl`, we can obtain the first few digits of the value of $K(k)$, for the selected value of $k$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import Elliptic: K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7548125779611365"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K(k^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Furthermore, using [`ArbFloats.jl`](https://github.com/JuliaArbTypes/ArbFloats.jl), we can obtain an extended precision evaluation of the elliptic integral $K$ (thanks to [@JeffreySarnoff](https://github.com/JeffreySarnoff) for this!):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ellipticK (generic function with 1 method)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using ArbFloats\n",
    "\n",
    "arb_precision = 270 # bits, select: 100..3200\n",
    "setprecision(ArbFloat, arb_precision)\n",
    "\n",
    "function ellipticK(x::ArbFloat)\n",
    "    arb_one    = one(ArbFloat)\n",
    "    arb_halfpi = ArbFloat(pi)/2\n",
    "        \n",
    "    k = sqrt( abs(x - arb_one) )\n",
    "    k = agm(arb_one, k)\n",
    "    k = arb_halfpi / k\n",
    "\n",
    "    return k\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.754812577961136589734955441226555912336722052661065250219997312395371169887377"
     ]
    }
   ],
   "source": [
    "showall(  ellipticK( (@ArbFloat 0.6051864057360395)^2 )  )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which differs from the result obtained by `TaylorIntegration.jl` __*only at the last digit !!!*__ Thus, `TaylorIntegration.jl` is able to evaluate special functions with *really* high accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 3. High-order polynomial approximations to special functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, what would happen if we wanted to evaluate the elliptic integral $K(k)$, not for one particular value, but for several values of the parameter $k$? Should we evaluate them one by one? If the set of values of $k$ lie within a small neighborhood of a given value of $k$, say $k_0$, then we can combine `TaylorSeries.jl` and `TaylorIntegration.jl` in order to obtain a high-order polynomial approximation to the special function $K$, around a given value $k_0$. Indeed, we may think of $k$ as a Taylor polynomial in $t$ ($t$ is a dummy variable), such that $k=k_0+t$, in order to obtain a polynomial representation of $K(k)$ for $k \\in (k_0-\\epsilon,k_0+\\epsilon)$ (where $\\epsilon$ is a small number)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using TaylorSeries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " 1.0 t + 𝒪(t¹⁶)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = Taylor1(15) # the dummy variable k around which we are expanding the parameter k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " 0.6051864057360395 + 1.0 t + 𝒪(t¹⁶)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = k0 + t # re-thinking k as a Taylor polynomial in the dummy variable t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now integrate an ODE system where the first equation corresponds to the original problem of evaluating the elliptic integral $K$, whereas the second equation corresponds to the parameter $k$, but since the parameter $k$ is constant wrt the integrating variable in the definition of $K$, we have $\\dot k = 0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "G! (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function G!(t, x, dx)\n",
    "    dx[1] = 1/sqrt(1-(x[2]*sin(t))^2)\n",
    "    dx[2] = zero(x[2])\n",
    "    nothing\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial conditions for this new system are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{TaylorSeries.Taylor1{Float64},1}:\n",
       "                         0.0 + 𝒪(t¹⁶)\n",
       "  0.6051864057360395 + 1.0 t + 𝒪(t¹⁶)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = [zero(t), k]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can integrate the system above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  1.147146 seconds (2.48 M allocations: 186.757 MiB, 4.43% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time tv, xv = taylorinteg(G!, x0, 0.0, π/2, 28, 1E-20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, `xv[end,1]` corresponds to a 15-th order Taylor expansion around $k_0$ of the complete elliptic integral of the first kind, $K$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " 1.7548125779611359 + 0.7902217547288612 t + 1.4862006794969997 t² + 2.4723818499390178 t³ + 4.877283348455082 t⁴ + 9.96559188808282 t⁵ + 21.224636460154077 t⁶ + 46.32399841153163 t⁷ + 103.09191906811044 t⁸ + 232.82644779824827 t⁹ + 532.0565841031579 t¹⁰ + 1227.5568728333137 t¹¹ + 2854.8206184572873 t¹² + 6683.860182924387 t¹³ + 15738.519788056736 t¹⁴ + 37243.53671901891 t¹⁵ + 𝒪(t¹⁶)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xv[end,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can exploit the function-like behavior (i.e., the \"callability\") of `Taylor1` variables in order to call this Taylor expansion of $K$ as if it were a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ellipticK_taylor (generic function with 1 method)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ellipticK_taylor(x) = xv[end,1](x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we're expanding around $k_0$, note that `ellipticK_taylor(x) ≈ Elliptic.K((x+k0)^2)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-6.661338147750939e-16"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ellipticK_taylor(0.0)-Elliptic.K(k0^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we will plot `ellipticK_taylor` and `Elliptic.K` for $k \\in [k_0-0.35,k_0+0.35]$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25518640573603957:0.025:0.9301864057360396"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_radius = 0.35\n",
    "x = k0-k_radius:0.025:k0+k_radius"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(x, ellipticK_taylor.(x-k0), lab=\"Taylor approx\")\n",
    "plot!(x, Elliptic.K.(x.^2), lab=\"Elliptic.K\")\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"Elliptic integral K value\")\n",
    "title!(\"Taylor polynomial approximation vs Elliptic.K\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot the absolute difference between `ellipticK_taylor` and `Elliptic.K` for $k$ in the same interval:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(x, log10.(abs.(ellipticK_taylor.(x-k0)-Elliptic.K.(x.^2))), lab=\"log10(abs(diff))\")\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"abs difference (log10)\")\n",
    "title!(\"K: Taylor approximation vs Elliptic.K\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we see that near $k_0$ the difference between the Taylor polynomial approximation and `Elliptic.K` is below $10^{-14}$; and at the bounds of the interval the difference grows to $10^{-3}$. We can obtain better approximations using higher-order Taylor polynomials, up to a point where `Float64` can no longer give better accuracies and extended precision should be used instead."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Furthermore, it is practically as efficient (if not even better!) to evaluate the elliptic integral $K$ with `Elliptic.jl` than it is to evaluate the Taylor polynomial approximation for it that we have obtained above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using BenchmarkTools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.49728786683491416"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k1 = k0+0.1\n",
    "k1sq = k1^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  268.143 ns (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.8517837163252886"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@btime Elliptic.K(k1sq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  114.540 ns (1 allocation: 16 bytes)\n"
     ]
    },
    {
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     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "@btime ellipticK_taylor(0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this approach is suitable to deal with more complicated or even custom-defined special functions!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. High-order polynomial approximations to integral transforms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section, we will try to obtain a high-order polynomial approximation to the Fourier transform of a smooth function $f$:\n",
    "\n",
    "$$\n",
    "\\hat f(\\omega) =\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^\\infty f(t)\\operatorname{e}^{-i\\omega t}\\mathrm{d}t\n",
    "$$\n",
    "\n",
    "We will deal with two cases:\n",
    "\n",
    "  + $f_1(t)=\\exp(-t^2)$\n",
    "  + $f_2(t)=\\cos(t)$ for $t \\in [-4\\pi,4\\pi]$; $0$ otherwise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "F2 (generic function with 1 method)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#f1 and f2:\n",
    "f1(t) = exp(-t^2)\n",
    "f2(t) = cos(t) #for t ∈ [−4π,4π]; 0 otherwise.\n",
    "\n",
    "#f1 and f2 Fourier transform defining ODEs:\n",
    "function F1!(t, x, dx)\n",
    "    dx[1] = f1(t)*exp(-im*x[2]*t)/(sqrt(2π))\n",
    "    dx[2] = zero(x[2])\n",
    "    nothing\n",
    "end\n",
    "\n",
    "function F2!(t, x, dx)\n",
    "    dx[1] = f2(t)*exp(-im*x[2]*t)/(sqrt(2π))\n",
    "    dx[2] = zero(x[2])\n",
    "    nothing\n",
    "end\n",
    "\n",
    "F1(ω) = exp(-(ω^2)/4)/sqrt(2) #analytical expression of Fourier transform of f1\n",
    "F2(ω) = sqrt(2/π)*ω*sin(4π*ω)/(ω^2-1) #analytical expression of Fourier transform of f1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\omega_0$ is the $\\omega$ value around which we will expand. That is, we will have $\\omega = \\omega_0 + \\delta \\omega$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 0.0im"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ω0 = 0.0+0im"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "  ( 1.0 ) t + 𝒪(t³¹)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "δω = Taylor1(Complex{Float64}, 30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "  ( 1.0 ) t + 𝒪(t³¹)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ω = ω0 + δω"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial condition for the ODE is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{TaylorSeries.Taylor1{Complex{Float64}},1}:\n",
       "  0.0 + 0.0im + 𝒪(t³¹)\n",
       "    ( 1.0 ) t + 𝒪(t³¹)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = [zero(δω), ω]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we perform the Taylor integration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  1.780823 seconds (4.23 M allocations: 746.953 MiB, 7.92% gc time)\n",
      "  0.843982 seconds (3.55 M allocations: 692.804 MiB, 13.15% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time tv1, xv1 = taylorinteg(F1!, x0, -10.0, 10.0, 28, 1E-20);\n",
    "@time tv2, xv2 = taylorinteg(F2!, x0, -4pi, 4pi, 28, 1E-20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We extract the corresponding real and imaginary parts of the Fourier transforms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "r1 = real(xv1[end,1])\n",
    "i1 = imag(xv1[end,1])\n",
    "\n",
    "r2 = real(xv2[end,1])\n",
    "i2 = imag(xv2[end,1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`y1` and `y2` are plotting auxiliaries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.99:0.02:0.99"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y1 = linspace(-4,4,100)\n",
    "y2 = linspace(-0.99,0.99,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the original functions and their Fourier transforms real and imaginary parts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "<polyline clip-path=\"url(#clip6400)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
       "  430.516,93.171 562.315,93.171 562.315,32.691 430.516,32.691 430.516,93.171 \n",
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       "<polyline clip-path=\"url(#clip6400)\" style=\"stroke:#009af9; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
       "  436.516,47.811 472.516,47.811 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip6400)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 478.516, 52.311)\" x=\"478.516\" y=\"52.311\">f(t) = exp(-t²)</text>\n",
       "</g>\n",
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       "  \"/>\n",
       "<g clip-path=\"url(#clip6400)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 478.516, 67.431)\" x=\"478.516\" y=\"67.431\">real(Fâ(?))</text>\n",
       "</g>\n",
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       "  436.516,78.051 472.516,78.051 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip6400)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 478.516, 82.551)\" x=\"478.516\" y=\"82.551\">imag(Fâ(?))</text>\n",
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       "</svg>\n"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# original f and Fourier(f), 1\n",
    "plot(y1, f1.(y1), lab=\"f(t) = exp(-t²)\")\n",
    "plot!(y1, r1.(y1), lab=\"real(F₁(ω))\")\n",
    "plot!(y1, i1.(y1), lab=\"imag(F₁(ω))\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 479.512, 52.311)\" x=\"479.512\" y=\"52.311\">fâ(t) = cos(t)</text>\n",
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     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# original f and Fourier(f), 2\n",
    "plot(y2, f2.(y2), lab=\"f₂(t) = cos(t)\")\n",
    "plot!(y2, r2.(y2), lab=\"real(F₂(ω))\")\n",
    "plot!(y2, i2.(y2), lab=\"imag(F₂(ω))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the analytical formulae of the corresponding Fourier transforms vs the integrated expressions obtained above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#integrated vs expected, 1\n",
    "plot(y1, r1.(y1), lab=\"integrated\")\n",
    "plot!(y1, F1.(y1), lab=\"analytical\")\n",
    "title!(\"f1 Fourier transform: analytical vs integrated\")\n",
    "xlabel!(\"omega\")\n",
    "ylabel!(\"real part of Fourier transf.\")"
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  },
  {
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   "execution_count": 48,
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     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#integrated vs expected, 2\n",
    "plot(y2, r2.(y2), lab=\"integrated\")\n",
    "plot!(y2, F2.(y2), lab=\"analytical\")\n",
    "title!(\"f2 Fourier transform: analytical vs integrated\")\n",
    "xlabel!(\"omega\")\n",
    "ylabel!(\"real part of Fourier transf.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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